Geometric Sequences

Suppose you were to draw an equilateral triangle on a sheet of paper. It might look something like this:

Now suppose that you draw lines connecting the midpoints of each of the edges of this triangle. This will dissect the larger triangle into four smaller triangles, each of which are equilateral. Three of these smaller triangles will be oriented in the same direction as the original triangle, whereas one will not. Consider the second image below, with the three triangles with the same orientation as the original triangle numbered.

We can continue to draw lines connecting the midpoints of the edges of the marked triangles and counting the resulting triangles that have the same orientation as the original triangle and we see that a pattern emerges.

What one notices is that each time we draw a new triangle by connecting the midpoints of the marked edges, we wind up with three times the number of triangles that were in the previous picture. So (assuming we had enough space) we could draw out the figure that would be the result of doing any number of these dissections. However, if we are only interested in knowing the number of triangles that each image will contain, we can take advantage of the fact that this pattern represents a geometric sequence.

A geometric sequence is a sequence with an initial term, a1 and a common ration, r, where each term after the initial term is obtained by multiplying the previous term by the ratio (a1 cannot be zero, and r cannot be zero or one).

In a geometric sequence, if we know the first term and the ratio, we can determine the nth term by the formula

an = a1*rn – 1

Similarly, if we know the first term and the ratio, we can determine the sum of the first n terms in a geometric sequence by the formula:

Sn =

a1(1 – rn)

1 – r

For the previous example with the triangles pointed in the same direction, we can show the results in the following table: